Optimal. Leaf size=202 \[ \frac{(a+b x)^{3/2} \sqrt{d+e x} (-a B e-4 A b e+5 b B d)}{2 e^2 (b d-a e)}-\frac{3 \sqrt{a+b x} \sqrt{d+e x} (-a B e-4 A b e+5 b B d)}{4 e^3}+\frac{3 (b d-a e) (-a B e-4 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 \sqrt{b} e^{7/2}}-\frac{2 (a+b x)^{5/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]
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Rubi [A] time = 0.166587, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {78, 50, 63, 217, 206} \[ \frac{(a+b x)^{3/2} \sqrt{d+e x} (-a B e-4 A b e+5 b B d)}{2 e^2 (b d-a e)}-\frac{3 \sqrt{a+b x} \sqrt{d+e x} (-a B e-4 A b e+5 b B d)}{4 e^3}+\frac{3 (b d-a e) (-a B e-4 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 \sqrt{b} e^{7/2}}-\frac{2 (a+b x)^{5/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 78
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt{d+e x}}+\frac{(5 b B d-4 A b e-a B e) \int \frac{(a+b x)^{3/2}}{\sqrt{d+e x}} \, dx}{e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt{d+e x}}+\frac{(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt{d+e x}}{2 e^2 (b d-a e)}-\frac{(3 (5 b B d-4 A b e-a B e)) \int \frac{\sqrt{a+b x}}{\sqrt{d+e x}} \, dx}{4 e^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt{d+e x}}-\frac{3 (5 b B d-4 A b e-a B e) \sqrt{a+b x} \sqrt{d+e x}}{4 e^3}+\frac{(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt{d+e x}}{2 e^2 (b d-a e)}+\frac{(3 (b d-a e) (5 b B d-4 A b e-a B e)) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{8 e^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt{d+e x}}-\frac{3 (5 b B d-4 A b e-a B e) \sqrt{a+b x} \sqrt{d+e x}}{4 e^3}+\frac{(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt{d+e x}}{2 e^2 (b d-a e)}+\frac{(3 (b d-a e) (5 b B d-4 A b e-a B e)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{a e}{b}+\frac{e x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{4 b e^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt{d+e x}}-\frac{3 (5 b B d-4 A b e-a B e) \sqrt{a+b x} \sqrt{d+e x}}{4 e^3}+\frac{(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt{d+e x}}{2 e^2 (b d-a e)}+\frac{(3 (b d-a e) (5 b B d-4 A b e-a B e)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{d+e x}}\right )}{4 b e^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt{d+e x}}-\frac{3 (5 b B d-4 A b e-a B e) \sqrt{a+b x} \sqrt{d+e x}}{4 e^3}+\frac{(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt{d+e x}}{2 e^2 (b d-a e)}+\frac{3 (b d-a e) (5 b B d-4 A b e-a B e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 \sqrt{b} e^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.605539, size = 168, normalized size = 0.83 \[ \frac{\sqrt{e} \sqrt{a+b x} \left (a e (-8 A e+13 B d+5 B e x)+4 A b e (3 d+e x)+b B \left (-15 d^2-5 d e x+2 e^2 x^2\right )\right )+\frac{3 (b d-a e)^{3/2} \sqrt{\frac{b (d+e x)}{b d-a e}} (-a B e-4 A b e+5 b B d) \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )}{b}}{4 e^{7/2} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 740, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.92089, size = 1281, normalized size = 6.34 \begin{align*} \left [\frac{3 \,{\left (5 \, B b^{2} d^{3} - 2 \,{\left (3 \, B a b + 2 \, A b^{2}\right )} d^{2} e +{\left (B a^{2} + 4 \, A a b\right )} d e^{2} +{\left (5 \, B b^{2} d^{2} e - 2 \,{\left (3 \, B a b + 2 \, A b^{2}\right )} d e^{2} +{\left (B a^{2} + 4 \, A a b\right )} e^{3}\right )} x\right )} \sqrt{b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b e x + b d + a e\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \,{\left (2 \, B b^{2} e^{3} x^{2} - 15 \, B b^{2} d^{2} e - 8 \, A a b e^{3} +{\left (13 \, B a b + 12 \, A b^{2}\right )} d e^{2} -{\left (5 \, B b^{2} d e^{2} -{\left (5 \, B a b + 4 \, A b^{2}\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{16 \,{\left (b e^{5} x + b d e^{4}\right )}}, -\frac{3 \,{\left (5 \, B b^{2} d^{3} - 2 \,{\left (3 \, B a b + 2 \, A b^{2}\right )} d^{2} e +{\left (B a^{2} + 4 \, A a b\right )} d e^{2} +{\left (5 \, B b^{2} d^{2} e - 2 \,{\left (3 \, B a b + 2 \, A b^{2}\right )} d e^{2} +{\left (B a^{2} + 4 \, A a b\right )} e^{3}\right )} x\right )} \sqrt{-b e} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d}}{2 \,{\left (b^{2} e^{2} x^{2} + a b d e +{\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \,{\left (2 \, B b^{2} e^{3} x^{2} - 15 \, B b^{2} d^{2} e - 8 \, A a b e^{3} +{\left (13 \, B a b + 12 \, A b^{2}\right )} d e^{2} -{\left (5 \, B b^{2} d e^{2} -{\left (5 \, B a b + 4 \, A b^{2}\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{8 \,{\left (b e^{5} x + b d e^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x\right ) \left (a + b x\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.55408, size = 367, normalized size = 1.82 \begin{align*} \frac{{\left ({\left (\frac{2 \,{\left (b x + a\right )} B b{\left | b \right |} e^{4}}{b^{8} d e^{6} - a b^{7} e^{7}} - \frac{5 \, B b^{2} d{\left | b \right |} e^{3} - B a b{\left | b \right |} e^{4} - 4 \, A b^{2}{\left | b \right |} e^{4}}{b^{8} d e^{6} - a b^{7} e^{7}}\right )}{\left (b x + a\right )} - \frac{3 \,{\left (5 \, B b^{3} d^{2}{\left | b \right |} e^{2} - 6 \, B a b^{2} d{\left | b \right |} e^{3} - 4 \, A b^{3} d{\left | b \right |} e^{3} + B a^{2} b{\left | b \right |} e^{4} + 4 \, A a b^{2}{\left | b \right |} e^{4}\right )}}{b^{8} d e^{6} - a b^{7} e^{7}}\right )} \sqrt{b x + a}}{1536 \, \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} - \frac{{\left (5 \, B b d{\left | b \right |} - B a{\left | b \right |} e - 4 \, A b{\left | b \right |} e\right )} e^{\left (-\frac{9}{2}\right )} \log \left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{512 \, b^{\frac{13}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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